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Properties

Label 2-1805-95.79-c0-0-0
Degree 22
Conductor 18051805
Sign 0.486−0.873i0.486 - 0.873i
Analytic cond. 0.9008120.900812
Root an. cond. 0.9491110.949111
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 − 0.909i)2-s + (−1.32 + 0.483i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (1.87 + 0.684i)6-s + (0.766 − 0.642i)9-s + (1.08 − 0.909i)10-s + (−0.707 − 1.22i)12-s + (1.32 + 0.483i)13-s + (−0.245 − 1.39i)15-s + (0.939 − 0.342i)16-s − 1.41·18-s − 1.00·20-s + (−0.939 − 0.342i)25-s + (−0.999 − 1.73i)26-s + ⋯
L(s)  = 1  + (−1.08 − 0.909i)2-s + (−1.32 + 0.483i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (1.87 + 0.684i)6-s + (0.766 − 0.642i)9-s + (1.08 − 0.909i)10-s + (−0.707 − 1.22i)12-s + (1.32 + 0.483i)13-s + (−0.245 − 1.39i)15-s + (0.939 − 0.342i)16-s − 1.41·18-s − 1.00·20-s + (−0.939 − 0.342i)25-s + (−0.999 − 1.73i)26-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((0.486−0.873i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s) L(s)=((0.486−0.873i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: 0.486−0.873i0.486 - 0.873i
Analytic conductor: 0.9008120.900812
Root analytic conductor: 0.9491110.949111
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1805(1029,⋅)\chi_{1805} (1029, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :0), 0.486−0.873i)(2,\ 1805,\ (\ :0),\ 0.486 - 0.873i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.33571497940.3357149794
L(12)L(\frac12) ≈\approx 0.33571497940.3357149794
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
19 1 1
good2 1+(1.08+0.909i)T+(0.173+0.984i)T2 1 + (1.08 + 0.909i)T + (0.173 + 0.984i)T^{2}
3 1+(1.32−0.483i)T+(0.766−0.642i)T2 1 + (1.32 - 0.483i)T + (0.766 - 0.642i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1+(−1.32−0.483i)T+(0.766+0.642i)T2 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2}
17 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
23 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1−1.41T+T2 1 - 1.41T + T^{2}
41 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
43 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
47 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
53 1+(−0.245−1.39i)T+(−0.939+0.342i)T2 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2}
59 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
61 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
67 1+(1.08−0.909i)T+(0.173−0.984i)T2 1 + (1.08 - 0.909i)T + (0.173 - 0.984i)T^{2}
71 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
73 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
79 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
97 1+(−1.08−0.909i)T+(0.173+0.984i)T2 1 + (-1.08 - 0.909i)T + (0.173 + 0.984i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.834456604049416608099916883478, −9.115742684208134934491106542606, −8.207416724356486193959675098830, −7.31530723543424235123443459576, −6.21225525464538832219178082596, −5.86880241614770173262660785248, −4.55244719803694623852970634402, −3.59357063301225749018263706735, −2.52610247492520836734286134394, −1.16145643419303792214584905934, 0.54542252911770121957812014451, 1.45888709436266364234402245954, 3.59982608745578708669701935652, 4.80347374695281852946417682883, 5.72703555347456897508434160732, 6.14631190221241046026892179166, 6.95153683815487225207121391669, 7.83839239677226577554220997837, 8.402340590831202336568180776200, 9.117253184464622146965083949953

Graph of the ZZ-function along the critical line